Eigenvalue problems in 𝐿^{∞}: optimality conditions, duality, and relations with optimal transport

In this article we characterize the L ∞ \mathrm {L}^\infty eigenvalue problem associated to the Rayleigh quotient ‖ ∇ u ‖ L ∞ / ‖ u ‖ ∞ \left .{\|\nabla u\|_{\mathrm {L}^\infty }}\middle /{\|u\|_\infty }\right . and relate it to a divergence-form PDE, similarly to what is known for L p \mathrm {L}^p eigenvalue problems and the p p -Laplacian for p > ∞ p>\infty . Contrary to existing methods, which study L ∞ \mathrm {L}^\infty -problems as limits of L p \mathrm {L}^p -problems for p → ∞ p\to \infty , we develop a novel framework for analyzing the limiting problem directly using convex analysis and geometric measure theory. For this, we derive a novel fine characterization of the subdifferential of the Lipschitz-constant-functional u ↦ ‖ ∇ u ‖ L ∞ u\mapsto \|\nabla u\|_{\mathrm {L}^\infty } . We show that the eigenvalue problem takes the form λ ν u = − div ⁡ ( τ ∇ τ u ) \lambda \nu u =-\operatorname {div}(\tau \nabla _\tau u) , where ν \nu and τ \tau are non-negative measures concentrated where | u | |u| respectively | ∇ u | |\nabla u| are maximal, and ∇ τ u \nabla _\tau u is the tangential gradient of u u